Three conditions. One declared metric.
The source subdivision map preserves boundaries, but norm preservation and generator compatibility are separate tests. A new conditional metric can satisfy all three while retaining the extra fine coordinates.
Inspect each transfer contract
Ordinary barycentric subdivision maps15 chain coordinates to149. Its degree squared norm factors are1,2,6,24. A universal √2 scaling does not normalize this map.
Generator defect values use the stated pairing. They are not endpoint-error bounds. The full proofs retain each branch's separate assumptions.
A coarse retraction supplies the missing structure
Let S subdivide an oriented simplex into its signed barycentric flags. Define R by mapping each new face-vertex to the average of its original vertices and extending through exterior products. Exactly, RS=I and RB′=BR. With Q=I−SR, every fine chain splits into S(Rx)+Qx.
H=RᵀGR+QᵀWQ, with positive G and W
The new squared norm is the old decoded norm plus the retained complement norm. It is positive definite and gives SᵀHS=G. Using the correct H-adjoint also gives A′H S=S A_G.
The old15-coordinate evolution is therefore preserved on its embedded image, at the same internal rate. The full149-coordinate operator still has134 complementary directions. This is an authored compatibility construction; the sources do not select its metric or coarse readout.
What the metric change costs
The metric generally has cross terms within each degree. Individual simplex squares no longer add to the total norm, so the original local stock/current rules cannot be carried over by name. A source law for locality, metric weights and measurement remains open.
The134 extra directions form a direct-sum chain complement. They are not134 qubits or a tensor donor bank. A complete transfer retains arbitrary complement information. Preparing zero complement amplitude, supplying the transfer clock and preserving outgoing records need their own implementation and resource accounting.
Exact singleton coefficients survive on the admitted coarse image; arbitrary fine states require a declared decoded observation. No free reset, indefinitely funded refinement, physical energy or spacetime identification follows.
Where √2 actually appears
It is the edge case of the factorial norm factors. The rectangle's rotated-halving fixed point, equal-amplitude two-channel normalization and the K+ carrier frequency √2 are separately typed statements. They supply no universal kinetic/potential coupling or saturation ceiling.
Next: derive the source rule selecting a metric/retraction and a local observable, or retain the original metric and explicitly account for an interface generator defect. The finite-window clock, memory, output and reset obligations remain in force.